Proposition 2 (Minimum Distance and Partial Distance [1]}) 

Let \(\mathcal {C}_0\)  be a \((n,k, d_0)\)  binary linear code. Let \(g\)  be an \(n\) -length vector with \(d_H(g,\mathcal {C}_0) = d_2\) . Let \(\mathcal {C}_1\)  be the \((n,k+1,d_1)\)  linear code obtained by adding the vector \(g\)  to \(\mathcal {C}_0\) , i.e., \(\mathcal {C}_1 = \langle g, \mathcal {C}_0 \rangle \) . Then \(d_ 2 = \min (d_0,d_1)\) .

